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International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019 On Prime Numbers and Related Applications V Yegnanarayanan, Veena Narayanan, R Srikanth Abstract: In this paper we probed some interesting aspects of Our motivating factor for this work stems from the classic primorial and factorial primes. We did some numerical analysis result of called Prime Number Theorem which beautifully about the distribution of prime numbers and tabulated our models the statistical behavioral pattern of huge primes viz., findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions chance for an arbitrarily picked 푛휖푁 to be a prime number is in control engineering and Brain networks. inverse in proportion to log(푛). Keywords: Factorial primes, Primorial primes, Prime numbers. II. RESULTS AND DISCUSSIONS I. INTRODUCTION A. Discussion on Primorial and Factorial Primes The irregular distribution of prime numbers makes it Our journey is driven by fascinating characteristics of difficult for us to presume and discover. Euclid about 2300 primordial primes and factorial primes. A primordial years before established that primes are infinitely many. Till function denoted 푃# is the product of all primes till P and date we are not aware of a formula for finding the nth 푛 includes P. The factorial function is 푛! = 푖=1 푖. A natural prime. It appears as if they are randomly embedded in N and curiosity pushes us to probe large primes. In order to do this, may be because of this that sequences of primes do not tend we need to get an idea by analyzing numbers such as to emanate from naturally happening processes. We 푛! + 1, 푛! − 1, 푃# + 1, 푃# − 1 . As these numbers have large constantly use computer encryption whenever we factors it is quite evident that their density of primes is more communicate our credit card information to Amazon or than the average. The following facts are well known. One logging into our bank account or posting an encrypted email. can also see [1]-[4] for more. It simply implies that we are using prime numbers and rely on 1 their properties for protection of life in this cyber-age. Carl (a) The density of primes close to M is log 푀. Sagan in his novel “Contact” remarked that prime numbers 푁 푁 are cool. He further said in that book that aliens prefer to (b) 푁! ≈ encode their message as long string of prime numbers. Many 푒 (c) If 푝휖 푁, 푁! + 1 , 푝 prime and 푁! + 1 ≢ of the cryptographic algorithms depend on finding large 0 (푚표푑 푝), then 푁! + 1 is a prime. prime numbers. So, if one did not perform the primality testing with a proper prime gap, then the algorithm itself The task of determination of huge primes comprises two would consume a very long time to run. Thus, one of the distinct steps. The first step is to determine the chance of interesting questions arising in the study of prime numbers is being a prime and the second step is testing for the primality. the distance between two successive primes. According to A number A is declared to have more chance of being a prime Grandville, understanding both the small and large prime if (푐퐴−1 − 1) ≡ 0(푚표푑 퐴). Here the option for c is not that gaps is extremely difficult and hence would not be answered pertinent except in typical situations where c is restricted until another era. from assuming certain forbidden values. It has already been Lately problems in number theory are handled with the estimated that the mathematical expectation of the number of math software tools such as maple or MATLAB. These tools 퐾2 2푑퐾 퐾2 primes from 퐾1 푡표 퐾2 is 퐼 = = 2 log( ) and hence processes complex computations with ease reducing 퐾1 퐾 퐾1 it takes significantly longer time to determine the succeeding voluminous labor. It is incredible to note that Goldbach’s factorial prime. Next, about the testing of primality, if a huge conjecture was found to be true through numerical techniques 5 for every number that is congruent to zero (mode 2) but less integer with more than 10 digits has higher chance of being a prime then in reality it mostly turns out to be so. A nice than or equal to 4 × 1018 . reference for practical tests for primality of huge numbers 5 with 10 or above digits is [5]. Starting with 2, as first prime number, it is customary to let 휋(푥) as number of prime less than or equal to x; 휗(푥) as the logarithm of product of all primes less than or equal to x, 훹(푥) as the logarithm of the lcm of all positive integers less than or equal to x; Also we let Revised Manuscript Received on October 05, 2019. 휋 푥 = 휗 푥 = 훹 푥 = 0 푖푓 푥 < 2 푎푛푑 푥휖푅 . In [6] the Veena Narayanan, Department of Mathematics, SASTRA Deemed authors have obtained nice upper and lower bounds for University, Thanjavur, India. Email: [email protected] V Yegnanarayanan, Department of Mathematics, SASTRA Deemed 휋 푥 , 휗 푥 푎푛푑 훹 푥 . University, Thanjavur, India. Email: [email protected] R Srikanth, TATA Realty-SASTRA Srinivasa Ramanujan Research chair for Number theory, SASTRA Deemed University, Thanjavur, India. Email: [email protected] Published By: Retrieval Number: L36521081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3652.1081219 4150 & Sciences Publication On Prime Numbers and Related Applications A natural approach to determine an approximation for the C. Numerical Analysis on Distribution on Primes 푥 푓(푦)푑푦 Here we provide a heuristic estimate for the primes of the summation 푓(푝) p ≤ x is to let it as equal to 2 . In log 푦 푘 푥 푑푦 푥 form 푦 = 2 푖=1 푥푖 + 1; 푥푖 ∈ 푃 − 2 ; 푥푖 ′푠 are distinct. If view of this, 휋 푥 푎푛푑 휗(푥) becomes 푎푛푑 푑푦 2 log 푦 2 π(n) = #{k ≤ n : k is prime} defines the number of primes not which is close to x. It has been universally agreed that the exceeding a fixed real number n, then the Prime Number 푛 following numerical approximations or estimations are Theorem (PNT) states that π n ~ where ln is the natural 푥 ln 푛 accurate and valid for almost all values. That is, log 푥 1 + logarithm. The Table I interprets this. 풏 12log푥<휋푥<푥log푥(1+32log푥) where 푥≥59 for lower Table-I: Calculation of 흅(풏)~ 푥 퐥퐧 풏 inequality and 푥 > 1 for upper inequality. 1 < 풏 log 푥 − n 흅(풏) 2 퐥퐧 풏 휋 푥 < 푥 for 푥 ≥ 67 for the lower inequality 10 4 4.3429 3 log 푥 − 2 50 15 12.7811 1.5 푥 and 푥 > 푒 = 4.48 for the upper inequality. log 푥 < 100 25 21.7142 500 95 80.4559 휋 푥 < 1.25506 푥 where 푥 ≥ 17 for lower inequality log 푥 1000 168 144.7648 1 and 푥 > 1 for the upper inequality. 푥 1 − 2 log 푥 < 5000 669 587.0478 휗 푥 < 푥 1 + 1 where 푥 ≥ 563 for the lower 2 log 푥 10000 1229 1085.7362 inequality and 푥 > 1 for the upper inequality. 1 − 푙표푔푥푒−푙표푔푥푅푥<휗(푥 ≤훹(푥) and 50000 5133 4621.16678 푙표푔 푥 − 휗 푥 ≤ 훹 푥 < 1 + 푙표푔 푥 푒 푅 푥 where 푅 = 100000 9592 8685.88964 515 Now define π*(n) = #{y ≤ n: y is prime} where y = 2x1x2 · ≈ 217.51631 and 푥 ≥ 2 for lower ( 546 − 322) · · xk + 1; xi ∈ P − {2} are distinct. Then the Table II gives the ratio of π*(n) to π (n). inequality and 푥 > 1 for the upper inequality. Table-II: . Ratio of π*(n) to π (n) Ratio n π(n) π*(n) 102 25 5 0.2 B. On Distribution of Primes 103 168 36 0.21428 It is a fact that till date there are 183 different proofs for the Euclid’s observation that primes are infinitely many. So by 104 1229 261 0.21236 beginning with 푝 = 2, if we presume that 푝 , 푝 , … , 푝 are 1 1 2 푘 From Table II it is clear that π*(n) ∼ α π (n) where α ≈ defined then by selecting a prime 푝 , one can continue the 푘+1 ∗ 푛 푘 0.21. Then by PNT we can say that 휋 (n)~α where α ≈ sequence for which N 푗 =1 푝푗 + 1 ≡ 0 푚표푑 푝푘+1 . Clearly ln 푛 0.21. Thus, we can enhance the approximation for the such a sequence of selection 푝푗 is not unique as there are property of being y to be prime as ln n. The Table III presents many options for 푝푘+1 to keep the above relation intact. Mullin [7] proposed two canonical options namely the least a comparison of the actual number to the estimated number of primes we considered. The Fig.1. represents the plot for 푝푘+1 and the greatest 푝푘+1. The former resulted in a sequence 2,3,7, 43, 13,53,5,6221671,38709183810571, Table III. 139,2081,11,17,5471, … [ 8] called first EM-sequence and Table-III: Comparison of Estimates the latter in 2,3,7,43,139,50207,340999, 2365347734339, … Actual Ratio n Expected [9] called the second EM-sequence. Mullin then asked number number whether these two sequences include in it every prime. In 102 5 4.56 1.096 [10] it was conjectured relaying on stochastic arguments that the second EM sequence would in all possibility could 103 36 30.4006 1.1841 exhaust all primes by having them in their length but till date not much is known about the first EM sequence.
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